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Distance-based bias in model-directed optimization of additively decomposable problems
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Distance-based bias in model-directed optimization of additively decomposable problems

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For many optimization problems it is possible to define a distance metric between problem variables that correlates with the likelihood and strength of interactions between the variables. For example, …

For many optimization problems it is possible to define a distance metric between problem variables that correlates with the likelihood and strength of interactions between the variables. For example, one may define a metric so that the dependencies between variables that are closer to each other with respect to the metric are expected to be stronger than the dependencies between variables that are further apart. The purpose of this paper is to describe a method that combines such a problem-specific distance metric with information mined from probabilistic models obtained in previous runs of estimation of distribution algorithms with the goal of solving future problem instances of similar type with increased speed, accuracy and reliability. While the focus of the paper is on additively decomposable problems and the hierarchical Bayesian optimization algorithm, it should be straightforward to generalize the approach to other model-directed optimization techniques and other problem classes. Compared to other techniques for learning from experience put forward in the past, the proposed technique is both more practical and more broadly applicable.

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  • 1. Distance-­‐Based  Bias     in  Model-­‐Directed  Op3miza3on    of  Addi3vely  Decomposable  Problems   Mar3n  Pelikan    and    Mark  W.  Hauschild     Missouri  Es3ma3on  of  Distribu3on  Algorithms  Laboratory   Department  of  Mathema3cs  and  Computer  Science   University  of  Missouri,  St.  Louis,  MO     E-­‐mail:  mar3n@mar3npelikan.net   WWW:  hKp://mar3npelikan.net/   1
  • 2. Background  •  Model-­‐directed  op3mizers  (MDOs)  learn  and  use   models  in  op3miza3on  to  solve  difficult   op3miza3on  problems  scalably  and  reliably.  •  MDOs  oPen  provide  more  than  the  solu3on;   they  provide  a  set  of  models  that  reveal   informa3on  about  the  problem.  •  Learning  from  experience:  Use  models  from  prior   runs  of  MDOs  to  introduce  bias  when  solving   problems  of  similar  type  in  future.   2
  • 3. Purpose  •  Combine  prior  models  with  a  problem-­‐specific   distance  metric  to  solve  new  problem  instances   with  increased  speed,  accuracy,  reliability.  •  Demonstrate  significant  speedups  across  a  broad   array  of  problem  domains.  •  Focus  on  hBOA  algorithm  and  addi3vely   decomposable  func3ons,  although  the  approach   can  be  generalized  to  other  MDOs  and  other   problem  classes.   3
  • 4. Outline  1.  Hierarchical  BOA  (hBOA).  2.  Distance  metric  for  ADFs.  3.  Learning  from  experience  via  distance-­‐based   bias.  4.  Experiments.  5.  Summary  and  conclusions.   4
  • 5. Hierarchical  Bayesian  Op3miza3on   Algorithm  (hBOA)   Current Bayesian Newpopulation Selection network population[Pelikan, Goldberg, & Cantu-Paz, 2001] 5
  • 6. Decision  Trees  Represent  Dependencies   Dependency X2   X1   X3   X4   Decision tree Probability table (more efficient) 6
  • 7. Learning  from  Experience     (Transfer  Learning)  •  Mo3va3on   –  When  solving  a  problem,  hBOA  provides  the  user   with  a  set  of  probabilis3c  models.   –  Each  model  encodes  informa3on  about  the  problem,   such  as  dependencies  between  variables.   –  Why  not  use  this  informa3on  when  solving  new   problem  instances  of  similar  type?  •  Example:  hBOA  solves  99  scheduling  problems;   why  not  use  the  knowledge  obtained  when   solving  the  100th  instance?   7
  • 8. How  to  Make  it  Work?  •  It  is  straighborward  to  keep  sta3s3cs  from  past   hBOA  runs,  for  example,  capturing  the  number  of   dependencies  between  any  pair  of  variables.  •  In  hBOA,  this  can  be  done  by  looking  at  the   number  of  “splits”  on  variable  Xi  in  a  decision  tree   storing  dependencies  for  variable  Xj.  •  But  it  is  important  to  ensure  that  the  sta3s3cs  are   meaningful  with  respect  to  the  problem  being   solved,  so  that  the  sta3s3cs  help  us  solve  future   problem  instances  faster  and  beKer.   8
  • 9. Learning  from  Experience  via   Distance-­‐Based  Bias:  Basic  Idea  •  Learning  from  experience  using  distance-­‐based  bias   –  Define  distances  between  problem  variables.   –  Mine  probabilis3c  models  from  previous  runs  for   model  regulari3es  with  respect  to  distances.  •  Mine  models  to  es3mate  how  strongly  variables   influence  each  other  depending  on  their  distance.   –  This  should  work  whenever  strength  of  dependencies   is  correlated  with  distance.  •  Apply  idea  to  hBOA  and  addi3vely  decomposable   func3ons.   9
  • 10. Addi3vely  Decomposable  Func3ons  •  Addi3vely  decomposable  func3on  (ADF):       –  {Si}  are  subsets  of  variables.   –  {fi}  are  func3ons  defining  overall  solu3on  quality.  •  Addi3vely  decomposable  func3ons  are  oPen   difficult  to  solve!  Many  NP-­‐complete  problems   are  ADFs  with  subproblems  of  2  or  3  variables.   10
  • 11. Define  Distance  Metric  for  ADFs   Using  Dependency  Graph  •  Create  a  dependency  graph  where  variables  in   the  same  subset  Si  are  connected.  •  Define  distance  between  variables  as  shortest   path  between  them  in  the  dependency  graph.  •  If  there  exists  no  such  path,  set  distance  to  the   number  of  variables  (any  exis3ng  path  is   shorter).  [Hauschild et al., 2008] 11
  • 12. Define  Distance  Metric  for  ADFs   Using  Dependency  Graph:  Example  [Hauschild et al., 2008] 12
  • 13. Mo3va3ng  Example  •  Propor3ons  of  splits  for  variables  at  various  distances   shows  evident  correla3on  between  the  two:   NK landscapes 2D spin glass 13
  • 14. Details  of  the  Approach  •  Denote  by  M  the  set  of  models  from  prior  runs.  •  Record  the  number  of  splits  on  any  variable  Xi  in   any  decision  tree  Xj  in  model  m  such  that  distance  of   Xi  and  Xj  is  d        •  Compute  probability  of  kth  split  on  variable  Xi  in  any   decision  tree  Xj  such  that  dist.  of  Xi  and  Xj  is  d   assuming  (k-­‐1)  such  splits:   14
  • 15. Details  of  the  Approach  •  Set  prior  probability  of  network  structure  based   on  the  learned  probabili3es  (kappa  denotes   strength  of  bias)  •  Evaluate  each  network  using  a  Bayesian  metric   15
  • 16. Test  Problems  •  Included  in  this  paper   –  NK  landscapes  with  nearest-­‐neighbor  interac3ons.   –  2D  spin  glass.  •  Done  later  on   –  3D  spin  glass.   –  Minimum  vertex  cover  for  random  graphs.   –  MAXSAT  for  3-­‐CNF  formulas.  •  Large  number  of  different  instances  for  each   problem  class  (100s  to  1000s  each).   16
  • 17. Experimental  Methodology  •  10-­‐fold  crossvalida3on   –  Divide  instances  into  10  sets.   –  Test  bias  from  models  on  9  sets  on  remaining  1  set,   repeat  for  every  set.   –  BoKom  line:  Any  problem  instance  is  never  used  for   both  crea3ng  the  bias  and  tes3ng  it.  •  Bisec3on  for  gemng  popula3on  sizes,  10  runs  for   each  problem  instance.  •  Focus  on  mul3plica3ve  speedups   –  How  many  3mes  faster  with  the  use  of  bias?   17
  • 18. Results  on  NK  Landscapes   18
  • 19. Results  on  Minimum  Vertex  Cover   19
  • 20. Results  on  2D  Spin  Glass   20
  • 21. Results  on  3D  Spin  Glass   21
  • 22. Results  on  MAXSAT   22
  • 23. More  Results  to  be  Published  Soon  •  Nearly  iden3cal  speedups  if  bias  is  based  on   problems  of  smaller  size.  •  Significant  speedups  even  if  bias  is  based  on   another  class  of  ADFs  (e.g.  models  from  NK   landscapes  used  to  solve  MVC).    •  Nearly  mul3plica3ve  speedups  in  combina3on   with  other  efficiency  enhancements  (e.g.  sporadic   model  building).  •  So  far  not  a  single  problem  class  for  which  the  bias   does  not  yield  significant  speedups.     23
  • 24. Results  Applicable  in  Other  Contexts  •  Approach  can  be  applied  to  other  model-­‐ directed  op3mizers,  such  as  ECGA,  LTGA,  or   mGA.  •  Approach  can  be  applied  to  other  problem   classes  for  which  a  distance  metric  can  be   defined,  such  as  QAP  or  scheduling  problems.  •  This  work  demonstrates  the  poten3al,  but  more   work  to  be  done  in  future.   24
  • 25. Summary  and  Conclusions  •  Proposed  a  prac3cal  approach  to  using  models   from  prior  runs  of  model-­‐directed  op3mizers  to   bias  op3miza3on  of  future  problem  instances.  •  Demonstrated  significant  speedups  across  a   number  of  problem  domains  and  semngs,   including  a  number  scenarios  that  are  not  possible   with  related  techniques  proposed  in  the  past.  •  Approach  is  ready  to  be  applied  in  a  different   context.   25
  • 26. Acknowledgments  •  Support  was  provided  by   –  NSF  grants  ECS-­‐0547013  and  IIS-­‐1115352.   –  ITS  at  the  University  of  Missouri  in  St.  Louis.   –  University  of  Missouri  Bioinforma3cs  Consor3um.  •  Get  the  papers  at   hKp://medal-­‐lab.org/files/2012001.pdf   hKp://medal-­‐lab.org/files/2012004.pdf   26

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